Question

Rút gọn biểu thức sau 

S = 1 + 1/3 + 1/3^2 + 1/3^3 + .......+ 1/3^n

Answer 1

\(S=1+\frac{1}{3}+\frac{1}{3^2}+.....+\frac{1}{3^n}\)

=>\(3S=3.\left(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^n}\right)=3+1+\frac{1}{3}+...+\frac{1}{3^{n-1}}\)

=>\(3S-S=\left(3+1+\frac{1}{3}+.....+\frac{1}{3^{n-1}}\right)-\left(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^n}\right)\)

=>\(2S=3+1+\frac{1}{3}+....+\frac{1}{3^{n-1}}-1-\frac{1}{3}-\frac{1}{3^2}-....-\frac{1}{3^n}=3-\frac{1}{3^n}=\frac{3^{n+1}-1}{3^n}\)

=>\(S=\frac{3^{n+1}-1}{3^n}:2=\frac{3^{n+1}-1}{3^n.2}\)

Vậy.................